Theorem 2nd0 4084

Description: The value of the second-member function at the empty set.

Assertion

Ref	Expression
2nd0	|- (2nd` (/)) = (/)

Proof of Theorem 2nd0

Step	Hyp	Ref	Expression
1		2ndval 4082	. 2 |- (2nd` (/)) = U.ran {(/)}
2		dmsn0 3324	. . . 4 |- dom {(/)} = (/)
3		dm0rn0 3330	. . . 4 |- (dom {(/)} = (/) <-> ran {(/)} = (/))
4	2, 3	mpbi 189	. . 3 |- ran {(/)} = (/)
5	4	unieqi 2511	. 2 |- U.ran {(/)} = U.(/)
6		uni0 2525	. 2 |- U.(/) = (/)
7	1, 5, 6	3eqtr 1499	1 |- (2nd` (/)) = (/)

Syntax hints:   = wceq 956  (/)c0 2280  {csn 2409  U.cuni 2503  dom cdm 3170  ran crn 3171  ` cfv 3182  2ndc2nd 4078

This theorem is referenced by:  smfval 8224  codval 10656  cmpval 10658

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-2nd 4080

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